Often observable data signals are composed of additive mixtures of unobservable source signals, one or more of which it would be useful to recover or remove from the data signal in a principled manner. By way of example, in many common signal transmission environments, multiple signal sources are active at the same time. (For instance, in the real world, many acoustic sources in the environment may be simultaneously generating sounds.) A receiver (such as a listener) often would like to attend to a single signal source, but any sensor (e.g. microphone) in the environment typically responds to a mixture of sources. As indicated schematically in FIG. 1, each component of such a sensor's response corresponds to some source, delayed by the propagation time between that source and the sensor, and further filtered by echoes, radiation characteristics of the sources, and so forth. We call such a component a sensor image of its source.
It has been considered useful to be able to recover the underlying source signals from the available response mixtures, so that a listener could listen to each signal source separately. This is the source separation problem.
In a very important and common version of the source separation problem, the radiated signals of the underlying sources are not observable in any way. That is to say, they cannot be detected, measured, or recorded in isolation. Rather, the only available relevant information is the response signals generated by the sensors (e.g. microphones) that are present in the environment. The signals from those sensors (the “response mixtures,” “sensor mixtures,” or simply “mixtures”) can be detected, captured, and processed.
From a signal processing perspective, the situation may be modeled as shown in FIG. 2. In this model it is assumed that the observable data signals m are composed of convolutive mixtures of the unknown sources s. The relationship between the hidden source signals and the data signals—the observable mixtures—are defined by a hidden “mixing matrix” H. An important signal processing challenge is to estimate those underlying but hidden sources by processing the observed sensor responses to create Source Images. This is referred to as the Blind Source Separation (BSS) problem.
For example, referring to FIG. 3A, a system 300 is shown that corresponds to a particular example of the more general system shown in FIG. 2. System 300 includes a plurality of sources 302a-c and a plurality of sensors 306a-c. Although the system 300 is shown as including four sources 302a-c and three sensors 306a-c, the particular numbers of sources and sensors shown in FIG. 3A is merely an example and does not constitute a limitation of the present invention, which may be used in connection with any number of sources and any number of sensors, and any number of mixture components. The number of sources need not, in general, be equal to the number of sensors.
The sources 302a-c emit corresponding signals 304a-c. More specifically, source 302a emits signal 304a, source 302b emits signal 304b, and source 302c emits signal 304c. Although in FIG. 3A each of the sources 302a-c is shown as emitting exactly one signal, this is merely an example and does not constitute a limitation of the present invention, which may be used in connection with sources that that emit any number of signals.
In FIG. 3A, each of the sensors 306a-c receives a mixture of one or more of the signals 304a-c. In practice, any particular sensor may receive zero, one, two, or more signals. In the particular example of FIG. 3A, sensor 306a receives a mixture of signals 304a and 304b; sensor 306b receives a mixture of signals 304b and 304c; and sensor 306c receives solely signal 304b. 
A signal source that contributes a mixture component with a statistically significant amount of energy to at least one sensor is called a contributing source. A source may be non-contributing either because it is inactive (not emitting a signal with any significant amount of energy, sometimes called being or becoming silent) or because its location in the environment, the signal propagation properties of the environment, and/or the location of the sensors in the environment combine to shield all sensors from its contribution. Additional factors that typically determine whether a particular source is contributing or not include the spectral content of the source signal, the transfer function of the environment, and the frequency response of the sensors.
The sensors 306a-c produce corresponding outputs 308a -c representing their input mixtures. These outputs are also called “responses” or “response signals.” For example, sensor 306a produces output 308a representing the mixture of signals 304a and 304b received by sensor 306a; sensor 306b produces output 308b representing the mixture of signals 304b and 304c received by sensor 306b; and sensor 306c produces output 308c representing the signal 304b received by sensor 306c. 
Although not specifically illustrated in FIG. 3A, the contribution that a particular signal makes to the mixture received by the sensors 306a-c may vary from source to source. For example, although in FIG. 3A both sensors 306a and 306b are shown as receiving signal 304b, properties of the signal 304b may in practice differ at sensors 306a and 306b, such as due to distances in distance traveled or other factors that dampen or otherwise modify the signal 304b on its way to sensors 306a and 306b. In many systems, there is a linear relationship between a source signal and the corresponding mixture component in the response of a particular sensor. This linear relationship can be described using a so-called “transfer function” that describes the propagation characteristics between the source and the sensor.
In many cases it would be advantageous to determine, or estimate, what each of the individual source signals 304a-c is. Techniques of processing sensor signals (which are mixtures) to separate sources from each other, are referred to as “Blind Source Separation” (BSS) algorithms. Here, the word “Blind” means that the only information available to the source separation system about the sources are the sensor responses—all of which are, in general, linear weighted mixtures of multiple sources. In other words, no “hidden” information about the sources themselves is available to the source separation system. The field of BSS processing is an active field of research—see, for instance, Aichner, et al (R. Aichner, H. Buchner, F. Yan, and W. Kellerman, “A real-time blind source separation scheme and its application to reverberant and noisy acoustic environments”, Signal Processing, vol. 86, pp. 1260-1277, 2007.) for a detailed description of a BSS algorithm.
As applied to FIG. 3A, for example, BSS may be used in an attempt to process the outputs 308a-c of sensors 306a-c, respectively, to identify the source signals 304a-c. For example, the system 300 of FIG. 3A includes a blind source separation module 310 which receives the signals 304a-c output by the sensors 306a-c and generates, based solely on those sensor outputs 308a-c, source identification outputs 312a-c which are intended to identify the source signals 304a-c that caused the sensors 306a-c to produce the outputs 308a-c. For example, the blind source separation module 310 may be used to process outputs 308a, 308b, and 308c (e.g., simultaneously) to produce outputs 312a-c, where output 312a is intended to estimate source signal 304a; output 312b is intended to estimate source signal 304b; and output 312c is intended to estimate source signal 304c. 
In the traditional BSS problem statement, the goal of the signal processing to be performed is to estimate the hidden source signals. However, this goal is itself problematic. In general, there are logical and mathematical limitations to what BSS algorithms can achieve. Note that the sources are truly hidden, and in general no pristine source signal is directly observable. Indeed, in many scenarios, including common acoustic environments, the very concept of a specific set of hidden source signals is ontologically suspect.
In these situations, the characterization of the sources as “hidden” actually masks a deeper problem: those signals are not well defined. This may be true, curiously enough, even though a BSS algorithm generates well-behaved estimates of the “hidden source signals.” This is possible because, in general, the power of an estimated source is different by an unknown amount from the power of the original hidden signal, the order of output estimates is typically unrelated to any particular enumeration of the input signals (the “permutation ambiguity”), the estimated source is time-shifted by an arbitrary amount relative to the original, and the spectral power profile of the estimate and its original is generally different. For this reason, each output of the BSS algorithm is referred to herein as a source image, building on the metaphoric understanding of images as being recognizable reproductions of some original (the hidden source signal), but differing in size, orientation, etc. A source image is any signal that is related to the putative hidden source signal of a particular signal source by a convolution kernel.
In fact, a source signal that is referred to as “hidden” actually is not any particular signal at all. Rather, it can best be considered to be an entire equivalence class of perfectly coherent signals. Thus, in a situation in which there are N simultaneously active sources, the computational situation can best be understood as a search for the definition of N equivalence classes, each Source Equivalence Class (SEC) corresponding to one of the active sources.
In this understanding of the BSS problem, each of the N estimated source images generated by the BSS algorithm is best understood as an estimate of some arbitrary member of one of the Source Equivalence Classes. Once any of the signals in an SEC is specified, all of the other signals in that class can, in theory, be generated, because any two signals in an SEC are related to each other via a finite length convolution kernel called an image kernel (and sometimes informally referred to as a “weight”). Given any member signal, there is another signal in the SEC corresponding to each possible convolution kernel.
Note that there are no image kernels capable of mapping a member of one SEC to a member of another SEC. This is because all members of one SEC are incoherent with all members of all other SECs. As a result, the expected value of a kernel defined by their ratios would have zero energy.
It will be understood that one of the members of each SEC might be regarded in some sense as the original “hidden source signal.” But that hidden member cannot, in general, be identified without imposing additional constraints on the computational problem. And, in many cases, the hidden source signal cannot be identified because it is, in the absence of any such defensible constraints, not well defined.
It is true that the hidden source member of the SEC can be defined, or at least a narrower subset of the SEC containing the hidden source member can be defined, if additional constraints are imposed by the physical situation or the statement of the problem to be solved. For example, if the physical locations of all of the signal sources and sensors are specified, the members of the SEC that might qualify as the original signal can be constrained. Much current work on the BSS problem takes the approach of attempting to better define the original source signal by imposing additional situational or computational constraints, and working through their computational consequences. Such systems are often identified as Blind Deconvolution and Blind System Identification systems.
A distinct and separate problem is to determine the component sensor images of each source. Note that, in general, none of the sensor images of a source will be identical to the corresponding hidden source signal. Nor will one of the sensor images of a source be identical to another image of the same source. Instead, each sensor image constitutes an independent view of its source. Because there are many signal processing systems that either require or can take advantage of multiple independent images of a signal source, particularly if each image can be associated with a specific sensor, it would be particularly advantageous to decompose every sensor signal into its constituent sensor images.